313edo: Difference between revisions

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{{Harmonics in equal|313}}
{{Harmonics in equal|313}}


=== Divisors ===
=== Subsets and supersets ===
313edo is the 65th [[prime edo]].
313edo is the 65th [[prime edo]].


Line 24: Line 24:
| 2.3
| 2.3
| {{monzo| -496 313 }}
| {{monzo| -496 313 }}
| [{{val| 313 496 }}]
| {{mapping| 313 496 }}
| +0.113
| +0.113
| 0.113
| 0.113
Line 31: Line 31:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| -1 49 -33 }}
| 32805/32768, {{monzo| -1 49 -33 }}
| [{{val| 313 496 727 }}]
| {{mapping| 313 496 727 }}
| -0.055
| -0.055
| 0.254
| 0.254
Line 38: Line 38:
| 2.3.5.7
| 2.3.5.7
| 6144/6125, 19683/19600, 40500000/40353607
| 6144/6125, 19683/19600, 40500000/40353607
| [{{val| 313 496 727 879 }}]
| {{mapping| 313 496 727 879 }}
| -0.143
| -0.143
| 0.268
| 0.268
Line 45: Line 45:
| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 5632/5625, 8019/8000, 43923/43904
| 540/539, 5632/5625, 8019/8000, 43923/43904
| [{{val| 313 496 727 879 1083 }}]
| {{mapping| 313 496 727 879 1083 }}
| -0.158
| -0.158
| 0.242
| 0.242
Line 52: Line 52:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 351/350, 540/539, 676/675, 4096/4095, 43923/43904
| 351/350, 540/539, 676/675, 4096/4095, 43923/43904
| [{{val| 313 496 727 879 1083 1158 }}]
| {{mapping| 313 496 727 879 1083 1158 }}
| -0.091
| -0.091
| 0.267
| 0.267
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
Line 137: Line 137:
| [[2/1]] (±0 ¢)
| [[2/1]] (±0 ¢)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Music ==
== Music ==
{{catrel|313edo tracks}}
{{Catrel| 313edo tracks }}


* [https://www.youtube.com/watch?v=ls51r_AICak Calling Voices] by [[User:Francium|Francium]]
; [[User:Francium|Francium]]
* "Calling Voices" from ''HemischisMatic EP'' – [https://open.spotify.com/track/6NdgC4jYz5TIvxV09kdpuk Spotify] | [https://francium223.bandcamp.com/track/calling-voices Bandcamp] | [https://youtu.be/ls51r_AICak?si=iaiiHo-Q-Q-74LN9 YouTube] – hemischis in 313edo tuning


[[Category:313edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Hemischis]]
[[Category:Hemischis]]
[[Category:Hera]]
[[Category:Hera]]
[[Category:Albus]]
[[Category:Albus]]

Revision as of 12:55, 21 October 2023

← 312edo 313edo 314edo →
Prime factorization 313 (prime)
Step size 3.83387 ¢ 
Fifth 183\313 (701.597 ¢)
Semitones (A1:m2) 29:24 (111.2 ¢ : 92.01 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

313edo provides the optimal patent val for 11- and 13-limit hemischis temperament and the 13-limit rank-3 temperaments madagascar and hera. It tempers out the schisma, 32805/32768, in the 5-limit; 6144/6125 and 19683/19600 in the 7-limit; 540/539, 5632/5625, 8019/8000 and 16384/16335 in the 11-limit; 351/350, 676/675, 729/728, 1001/1000, 2080/2079 and 4096/4095 in the 13-limit.

Prime harmonics

Approximation of prime harmonics in 313edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.36 +0.91 +1.14 +0.76 -0.91 -1.44 +1.53 +0.48 +1.73 +1.29
Relative (%) +0.0 -9.3 +23.7 +29.8 +19.8 -23.8 -37.6 +39.9 +12.5 +45.2 +33.7
Steps
(reduced) 		313
(0) 		496
(183) 		727
(101) 		879
(253) 		1083
(144) 		1158
(219) 		1279
(27) 		1330
(78) 		1416
(164) 		1521
(269) 		1551
(299)

Subsets and supersets

313edo is the 65th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-496 313⟩ [⟨313 496]] +0.113 0.113 2.94
2.3.5 32805/32768, [-1 49 -33⟩ [⟨313 496 727]] -0.055 0.254 6.64
2.3.5.7 6144/6125, 19683/19600, 40500000/40353607 [⟨313 496 727 879]] -0.143 0.268 6.99
2.3.5.7.11 540/539, 5632/5625, 8019/8000, 43923/43904 [⟨313 496 727 879 1083]] -0.158 0.242 6.30
2.3.5.7.11.13 351/350, 540/539, 676/675, 4096/4095, 43923/43904 [⟨313 496 727 879 1083 1158]] -0.091 0.267 6.97

Rank-2 temperaments

Periods
per 8ve 		Generator* Cents* Associated
Ratio 		Temperaments
1 26\313 99.68 18/17 Quintaschis
1 65\313 249.20 15/13 Hemischis
1 130\313 498.40 4/3 Helmholtz
1 156\313 598.08 847/600 Vydubychi

Scales

Madagascar[9] (or Barbados[9]) scale
Step Cents JI Interpretation
53 (53\313) 203.195 9/8 (-0.715 ¢)
12 (65\313) 249.201 15/13 (+1.46 ¢)
53 (118\313) 452.396 13/10 (-1,818 ¢)
12 (130\313) 498.403 4/3 (+0.358 ¢)
53 (183\313) 701.597 3/2 (-0.358 ¢)
12 (195\313) 747.604 20/13 (+1.818 ¢)
53 (248\313) 950.799 26/15 (-1.46 ¢)
12 (260\313) 996.805 16/9 (+0.715 ¢)
53 (313\313) 1200.000 2/1 (±0 ¢)

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

See also: Category:313edo tracks
Francium
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